# Null

From MASS v7.3-40
by Brian Ripley

##### Null Spaces of Matrices

Given a matrix, `M`

, find a matrix `N`

giving a basis for the
(left) null space. That is `crossprod(N, M) = t(N) %*% M`

is an all-zero matrix and `N`

has the maximum number of linearly
independent columns.

- Keywords
- algebra

##### Usage

`Null(M)`

##### Arguments

- M
- Input matrix. A vector is coerced to a 1-column matrix.

##### Details

For a basis for the (right) null space
${x : Mx = 0}$,
use `Null(t(M))`

.

##### Value

- The matrix
`N`

with the basis for the (left) null space, or a matrix with zero columns if the matrix`M`

is square and of maximal rank.

##### References

Venables, W. N. and Ripley, B. D. (2002)
*Modern Applied Statistics with S.* Fourth edition. Springer.

##### See Also

##### Examples

```
# The function is currently defined as
function(M)
{
tmp <- qr(M)
set <- if(tmp$rank == 0L) seq_len(ncol(M)) else -seq_len(tmp$rank)
qr.Q(tmp, complete = TRUE)[, set, drop = FALSE]
}
```

*Documentation reproduced from package MASS, version 7.3-40, License: GPL-2 | GPL-3*

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